@article{article-minimal,
    author = "Leslie B. Lamport",
    title = "blah blah blah",
    journal = "Some outlet",
    year = "1986"
}

@Article{pysal2007,
    author = "Rey, Sergio J. and Anselin, Luc",
    title = "{PySAL: A Python Library of Spatial Analytical Methods}",
    journal = "The Review of Regional Studies",
    year = "2007",
    volume = "37",
    number = "1",
    pages = "5-27",
    keywords = "Open Source; Software; Spatial"
}

@Article{Lee2001,
    author = "Lee, Sang-Il",
    title = "Developing a bivariate spatial association measure: An integration of {Pearson's r and Moran's I}",
    journal = "Journal of Geographical Systems",
    year = "2001",
    month = "Dec",
    day = "01",
    volume = "3",
    number = "4",
    pages = "369--385",
    abstract = "{\enspace}This research is concerned with developing a bivariate spatial association measure or spatial correlation coefficient, which is intended to capture spatial association among observations in terms of their point-to-point relationships across two spatial patterns. The need for parameterization of the bivariate spatial dependence is precipitated by the realization that aspatial bivariate association measures, such as Pearson's correlation coefficient, do not recognize spatial distributional aspects of data sets. This study devises an L statistic by integrating Pearson's r as an aspatial bivariate association measure and Moran's I as a univariate spatial association measure. The concept of a spatial smoothing scalar (SSS) plays a pivotal role in this task.",
    issn = "1435-5930",
    doi = "10.1007/s101090100064",
    url = "https://doi.org/10.1007/s101090100064"
}

@article{Benjamini:2001,
    Author = "Benjamini, Yoav and Yekutieli, Daniel",
    Abstract = "Benjamini and Hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems. A simple procedure was given there as an FDR controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate. We prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses. This condition for positive dependency is general enough to cover many problems of practical interest, including the comparisons of many treatments with a single control, multivariate normal test statistics with positive correlation matrix and multivariate t. Furthermore, the test statistics may be discrete, and the tested hypotheses composite without posing special difficulties. For all other forms of dependency, a simple conservative modification of the procedure controls the false discovery rate. Thus the range of problems for which a procedure with proven FDR control can be offered is greatly increased.",
    Date-Added = "2019-03-28 22:46:12 -0700",
    Date-Modified = "2019-03-28 22:46:28 -0700",
    Issn = "00905364",
    Journal = "The Annals of Statistics",
    Number = "4",
    Pages = "1165--1188",
    Publisher = "Institute of Mathematical Statistics",
    Title = "The Control of the False Discovery Rate in Multiple Testing under Dependency",
    Url = "http://www.jstor.org/stable/2674075",
    Volume = "29",
    Year = "2001",
    Bdsk-File-1 = " ",
    Bdsk-Url-1 = "http://www.jstor.org/stable/2674075"
}

@article{Castro:2006tz,
    Author = "de Castro, Marcia Caldas and Singer, Burton H.",
    Citeulike-Article-Id = "647229",
    Date-Added = "2019-03-28 22:46:02 -0700",
    Date-Modified = "2019-03-28 22:46:02 -0700",
    Doi = "10.1111/j.0016-7363.2006.00682.x",
    Issn = "0016-7363",
    Journal = "Geographical Analysis",
    Keywords = "local statistics, multiple comparisons, false discovery rate",
    Month = "April",
    Number = "2",
    Pages = "180-208",
    Publisher = "Blackwell Publishing",
    Title = "Controlling the False Discovery Rate: A New Application to Account for Multiple and Dependent Tests in Local Statistics of Spatial Association",
    Url = "http://dx.doi.org/10.1111/j.0016-7363.2006.00682.x",
    Volume = "38",
    Year = "2006",
    Bdsk-File-1 = " ",
    Bdsk-Url-1 = "http://dx.doi.org/10.1111/j.0016-7363.2006.00682.x"
}

@article{Anselin95,
    Author = "Anselin, Luc",
    Doi = "10.1111/j.1538-4632.1995.tb00338.x",
    Issn = "0016-7363",
    Journal = "Geographical Analysis",
    Month = "Sep",
    Number = "2",
    Pages = "93--115",
    Publisher = "Wiley",
    Title = "Local Indicators of Spatial Association-{LISA}",
    Url = "http://dx.doi.org/10.1111/j.1538-4632.1995.tb00338.x",
    Volume = "27",
    Year = "1995",
    Bdsk-Url-1 = "http://dx.doi.org/10.1111/j.1538-4632.1995.tb00338.x"
}

@article{Assun_o_1999,
    Author = "Assuncao, Renato M. and Reis, Edna A.",
    Doi = "10.1002/(sici)1097-0258(19990830)18:16<2147::aid-sim179>3.0.co;2-i",
    Issn = "1097-0258",
    Journal = "Statistics in Medicine",
    Month = "Aug",
    Number = "16",
    Pages = "2147--2162",
    Publisher = "Wiley",
    Title = "A new proposal to adjust {Moran's I} for population density",
    Url = "http://dx.doi.org/10.1002/(sici)1097-0258(19990830)18:16<2147::aid-sim179>3.0.co;2-i",
    Volume = "18",
    Year = "1999",
    Bdsk-Url-1 = "http://dx.doi.org/10.1002/(sici)1097-0258(19990830)18:16\%3C2147::aid-sim179\%3E3.0.co;2-i"
}

@book{cliff81,
    Author = "Cliff, A.D. and Ord, J.K.",
    Address = "London",
    Publisher = "Pion",
    Title = "Spatial Processes: Models and Applications",
    Year = "1981"
}

@article{duque18,
    author = "Duque, Juan C. and Laniado, H. and Polo, A.",
    journal = "PLOS ONE",
    publisher = "Public Library of Science",
    title = "S-maup: Statistical Test to Measure the Sensitivity to the Modifiable Areal Unit Problem",
    year = "2018",
    month = "11",
    volume = "13",
    url = "https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0207377",
    pages = "1-25",
    abstract = "This work presents a nonparametric statistical test, S-maup, to measure the sensitivity of a spatially intensive variable to the effects of the Modifiable Areal Unit Problem (MAUP). To the best of our knowledge, S-maup is the first statistic of its type and focuses on determining how much the distribution of the variable, at its highest level of spatial disaggregation, will change when it is spatially aggregated. Through a computational experiment, we obtain the basis for the design of the statistical test under the null hypothesis of non-sensitivity to MAUP. We performed an exhaustive simulation study for approaching the empirical distribution of the statistical test, obtaining its critical values, and computing its power and size. The results indicate that, in general, both the statistical size and power improve with increasing sample size. Finally, for illustrative purposes, an empirical application is made using the Mincer equation in South Africa, where starting from 206 municipalities, the S-maup statistic is used to find the maximum level of spatial aggregation that avoids the negative consequences of the MAUP.",
    number = "11",
    doi = "https://doi.org/10.1371/journal.pone.0207377"
}

@Article{Getis_2010,
    author = "Getis, Arthur and Ord, J. K.",
    title = "The Analysis of Spatial Association by Use of Distance Statistics",
    year = "2010",
    volume = "24",
    number = "3",
    month = "Sep",
    pages = "189–206",
    issn = "0016-7363",
    doi = "10.1111/j.1538-4632.1992.tb00261.x",
    url = "http://dx.doi.org/10.1111/j.1538-4632.1992.tb00261.x",
    journal = "Geographical Analysis",
    publisher = "Wiley"
}

@article{Ord_2010,
    Author = "Ord, J. K. and Getis, Arthur",
    Doi = "10.1111/j.1538-4632.1995.tb00912.x",
    Issn = "0016-7363",
    Journal = "Geographical Analysis",
    Month = "Sep",
    Number = "4",
    Pages = "286--306",
    Publisher = "Wiley",
    Title = "Local Spatial Autocorrelation Statistics: Distributional Issues and an Application",
    Url = "http://dx.doi.org/10.1111/j.1538-4632.1995.tb00912.x",
    Volume = "27",
    Year = "2010",
    Bdsk-Url-1 = "http://dx.doi.org/10.1111/j.1538-4632.1995.tb00912.x"
}

@article{Hubert_1981,
    Author = "Hubert, L. J. and Golledge, R. G. and Costanzo, C. M.",
    Doi = "10.1111/j.1538-4632.1981.tb00731.x",
    Issn = "0016-7363",
    Journal = "Geographical Analysis",
    Month = "Sep",
    Number = "3",
    Pages = "224--233",
    Publisher = "Wiley",
    Title = "Generalized Procedures for Evaluating Spatial Autocorrelation",
    Url = "http://dx.doi.org/10.1111/j.1538-4632.1981.tb00731.x",
    Volume = "13",
    Year = "1981",
    Bdsk-Url-1 = "http://dx.doi.org/10.1111/j.1538-4632.1981.tb00731.x"
}

@article{Kang2018,
    Author = "Kang, Wei and Rey, Sergio J.",
    Abstract = "Spatial effects have been recognized to play an important role in transitional dynamics of regional incomes. Detection and evaluation of both spatial heterogeneity and spatial dependence in discrete Markov chain models, which have been widely applied to the study of regional income distribution dynamics and convergence, are vital, but under-explored issues. Indeed, in this spatiotemporal setting, spatial effects can take much more complex forms than that in a pure cross-sectional setting. In this paper, we address two test frameworks. The first is a conditional spatial Markov chains test framework, which can be used to detect spatial heterogeneity and temporally lagged spatial dependence; the second is a joint spatial Markov chains test framework, which tests for contemporaneous spatial dependence. A series of Monte Carlo experiments are designed to examine size, power and robustness properties of these tests for a range of sample sizes (spatial {\$}{\$}{\backslash}times {\$}{\$}{\texttimes}temporal dimensions), for different levels of discretization granularity and for different number of regimes. Results indicate that all tests display good size property except when sample size is fairly small. All tests for spatial dependence are similar in almost all aspects---size, power and robustness. Conditional spatial Markov tests for spatial heterogeneity have highest power for detecting spatial heterogeneity. Granularity of discretization has a major impact on the size properties of the tests when sample size is fairly small.",
    Date-Added = "2018-12-24 21:05:56 -0800",
    Date-Modified = "2018-12-24 21:05:56 -0800",
    Day = "01",
    Doi = "10.1007/s00168-017-0859-9",
    Issn = "1432-0592",
    Journal = "The Annals of Regional Science",
    Month = "Jul",
    Number = "1",
    Pages = "73--93",
    Title = "Conditional and joint tests for spatial effects in discrete Markov chain models of regional income distribution dynamics",
    Url = "https://doi.org/10.1007/s00168-017-0859-9",
    Volume = "61",
    Year = "2018",
    Bdsk-Url-1 = "https://doi.org/10.1007/s00168-017-0859-9"
}

@book{Ibe2009,
    Author = "Ibe, Oliver",
    Address = "Amsterdam",
    Date-Added = "2018-09-10 17:28:24 -0700",
    Date-Modified = "2018-09-10 17:38:32 -0700",
    Keywords = "markov",
    Publisher = "Elsevier Academic Press",
    Title = "Markov processes for stochastic modeling",
    Year = "2009"
}

@article{Rey2014a,
    Author = "Rey, Sergio J.",
    Date-Added = "2018-09-10 14:49:27 -0700",
    Date-Modified = "2018-09-10 14:50:05 -0700",
    Journal = "Journal of Geographical Systems",
    Keywords = "markov, space-time",
    Number = "2",
    Pages = "115-137",
    Title = "Rank-based {Markov} chains for regional income distribution dynamics",
    Volume = "16",
    Year = "2014"
}

@article{Formby:2004fk,
    Author = "Formby, John P. and Smith, W. James and Zheng, Buhong",
    Date-Added = "2018-07-06 22:21:00 +0000",
    Date-Modified = "2018-07-06 22:21:00 +0000",
    Doi = "https://doi.org/10.1016/S0304-4076(03)00211-2",
    Journal = "Journal of Econometrics",
    Keywords = "mobility, distributional dynamics",
    Number = "1",
    Pages = "181--205",
    Publisher = "Elsevier",
    Title = "Mobility measurement, transition matrices and statistical inference",
    Url = "http://www.sciencedirect.com/science/article/pii/S0304407603002112",
    Volume = "120",
    Year = "2004",
    Bdsk-Url-1 = "http://www.sciencedirect.com/science/article/pii/S0304407603002112",
    Bdsk-Url-2 = "https://doi.org/10.1016/S0304-4076(03)00211-2"
}

@article{Rey2011,
    Author = "Rey, Sergio J. and Murray, Alan T. and Anselin, Luc",
    Date-Added = "2018-07-03 17:53:40 +0000",
    Date-Modified = "2018-07-03 17:53:45 +0000",
    Doi = "10.1007/s12076-010-0048-2",
    Journal = "Letters in Spatial and Resource Sciences",
    Keywords = "directional.py",
    Number = "1",
    Pages = "81--90",
    Publisher = "Springer",
    Title = "Visualizing regional income distribution dynamics",
    Url = "https://doi.org/10.1007/s12076-010-0048-2",
    Volume = "4",
    Year = "2011",
    Bdsk-Url-1 = "https://doi.org/10.1007/s12076-010-0048-2"
}

@article{Rey2016a,
    Author = "Rey, Sergio J. and Kang, Wei and Wolf, Levi",
    Abstract = "Discrete Markov chain models (DMCs) have been widely applied to the study of regional income distribution dynamics and convergence. This popularity reflects the rich body of DMC theory on the one hand and the ability of this framework to provide insights on the internal and external properties of regional income distribution dynamics on the other. In this paper we examine the properties of tests for spatial effects in DMC models of regional distribution dynamics. We do so through a series of Monte Carlo simulations designed to examine the size, power and robustness of tests for spatial heterogeneity and spatial dependence in transitional dynamics. This requires that we specify a data generating process for not only the null, but also alternatives when spatial heterogeneity or spatial dependence is present in the transitional dynamics. We are not aware of any work which has examined these types of data generating processes in the spatial distribution dynamics literature. Results indicate that tests for spatial heterogeneity and spatial dependence display good power for the presence of spatial effects. However, tests for spatial heterogeneity are not robust to the presence of strong spatial dependence, while tests for spatial dependence are sensitive to the spatial configuration of heterogeneity. When the spatial configuration can be considered random, dependence tests are robust to the dynamic spatial heterogeneity, but not so to the process mean heterogeneity when the difference in process means is large relative to the variance of the time series.",
    Date-Added = "2018-06-20 17:13:29 +0000",
    Date-Modified = "2018-06-20 17:13:38 +0000",
    Doi = "10.1007/s10109-016-0234-x",
    Issn = "1435-5949",
    Journal = "Journal of Geographical Systems",
    Number = "4",
    Pages = "377--398",
    Title = "The properties of tests for spatial effects in discrete {Markov} chain models of regional income distribution dynamics",
    Url = "http://dx.doi.org/10.1007/s10109-016-0234-x",
    Volume = "18",
    Year = "2016",
    Bdsk-Url-1 = "http://dx.doi.org/10.1007/s10109-016-0234-x"
}

@article{Rey2016,
    Author = "Rey, Sergio J.",
    Abstract = "In the study of income inequality dynamics, the concept of exchange mobility plays a central role. Applications of classical rank correlation statistics have been used to assess the degree to which individual economies swap positions in the income distribution over time. These classic measures ignore the underlying geographical pattern of rank changes. Rey (2004) introduced a spatial concordance statistic as an extension of Kendall's rank correlation statistic, a commonly employed measure of exchange mobility. This article suggests local forms of the global spatial concordance statistic: local indicators of mobility association (LIMA). The LIMA statistics allow for the decomposition of the global measure into the contributions associated with individual locations. They do so by considering the degree of concordance (stability) or discordance (exchange mobility) reflected within an economy's local spatial context. Different forms of the LIMAs derive from alternative expressions of the neighborhood and neighbor set. Additionally, the additive decomposition of the LIMAs permits the development of a mesolevel analytic to examine whether the overall space--time concordance is driven by either interregional or intraregional concordance. The measures are illustrated in a case study that examines regional income dynamics in Mexico.",
    Date-Added = "2018-06-12 22:59:18 +0000",
    Date-Modified = "2018-06-12 22:59:33 +0000",
    Doi = "10.1080/24694452.2016.1151336",
    Journal = "Annals of the American Association of Geographers",
    Keywords = "LIMA",
    Number = "4",
    Pages = "788-803",
    Title = "Space--Time Patterns of Rank Concordance: Local Indicators of Mobility Association with Application to Spatial Income Inequality Dynamics",
    Url = "https://doi.org/10.1080/24694452.2016.1151336",
    Volume = "106",
    Year = "2016",
    Bdsk-Url-1 = "https://doi.org/10.1080/24694452.2016.1151336"
}

@incollection{Rey2004a,
    Author = "Rey, Sergio J.",
    Editor = "Getis, A. and M{\'{u}}r, J. and Zoeller, H.",
    Abstract = "This chapter discusses the importance of distributional dynamics in the evolution of regional incomes. Serge starts with spatial Markov stuff, but then extends it to include the spatial tau decomposition and the measures of inter-intra regional cohesion.",
    Address = "Hampshire",
    Booktitle = "Spatial econometrics and spatial statistics",
    Date-Added = "2018-06-12 22:57:36 +0000",
    Date-Modified = "2018-06-12 22:57:51 +0000",
    Keywords = "convergence, rank.py",
    Pages = "193-213",
    Publisher = "Palgrave",
    Title = "Spatial dependence in the evolution of regional income distributions",
    Year = "2004"
}

@article{Rey2014,
    Author = "Rey, Sergio J.",
    Date-Added = "2018-06-12 22:54:57 +0000",
    Date-Modified = "2018-06-12 22:55:04 +0000",
    Doi = "10.1007/s00180-013-0461-2",
    Journal = "Computational Statistics",
    Keywords = "concordance, rank.py, LIMA",
    Number = "3-4",
    Pages = "799-811",
    Publisher = "Springer",
    Title = "Fast algorithms for a space-time concordance measure",
    Url = "https://doi.org/10.1007/s00180-013-0461-2",
    Volume = "29",
    Year = "2014",
    Bdsk-Url-1 = "https://doi.org/10.1007/s00180-013-0461-2"
}

@book{Press2007,
    Author = "Press, William H and Teukolsky, Saul A and Vetterling, William T and Flannery, Brian P",
    Address = "Cambridge",
    Date-Added = "2018-06-12 22:53:57 +0000",
    Date-Modified = "2018-06-12 22:54:05 +0000",
    Edition = "3rd",
    Keywords = "rank.py",
    Publisher = "Cambridge Univ Pr",
    Title = "Numerical recipes: the art of scientific computing",
    Year = "2007"
}

@article{Christensen2005,
    Author = "Christensen, David",
    Abstract = "Traditional algorithms for the calculation of Kendall's $\tau$ between two datasets of n samples have a calculation time of O(n2). This paper presents a suite of algorithms with expected calculation time of O(n log n) or better using a combination of sorting and balanced tree data structures. The literature, e.g. Dwork et al. (2001), has alluded to the existence of O(n log n) algorithms without any analysis: this paper gives an explicit descriptions of such algorithms for general use both for the case with and without duplicate values in the data. Execution times for sample data are reduced from 3.8 hours to around 1--2 seconds for one million data pairs.",
    Date-Added = "2018-06-12 22:50:55 +0000",
    Date-Modified = "2018-06-12 22:51:11 +0000",
    Doi = "10.1007/BF02736122",
    Journal = "Computational Statistics",
    Month = "Mar",
    Number = "1",
    Pages = "51--62",
    Title = "Fast algorithms for the calculation of Kendall's $\tau$",
    Url = {https://doi.org/10.1007/BF02736122"},
    Volume = "20",
    Year = "2005",
    Bdsk-Url-1 = "https://doi.org/10.1007/BF02736122\%22",
    Bdsk-Url-2 = "https://doi.org/10.1007/BF02736122"
}

@article{Kullback1962,
    Author = "Kullback, S. and Kupperman, M. and Ku, H. H.",
    Date-Added = "2018-06-12 22:49:01 +0000",
    Date-Modified = "2018-06-12 22:49:07 +0000",
    Doi = "10.2307/1266291",
    Journal = "Technometrics",
    Keywords = "markov, markov.py",
    Number = "4",
    Pages = "573--608",
    Publisher = "JSTOR",
    Title = "Tests for contingency tables and {Markov} chains",
    Url = "http://www.jstor.org/stable/1266291",
    Volume = "4",
    Year = "1962",
    Bdsk-Url-1 = "http://www.jstor.org/stable/1266291",
    Bdsk-Url-2 = "https://doi.org/10.2307/1266291"
}

@article{Bickenbach2003,
    Author = "Bickenbach, Frank and Bode, Eckhardt",
    Date-Added = "2018-06-12 22:47:20 +0000",
    Date-Modified = "2018-06-12 22:47:31 +0000",
    Doi = "10.1177/0160017603253789",
    Journal = "International Regional Science Review",
    Number = "3",
    Pages = "363-392",
    Title = "Evaluating the {Markov} property in studies of economic convergence",
    Url = "https://doi.org/10.1177/0160017603253789",
    Volume = "26",
    Year = "2003",
    Bdsk-Url-1 = "https://doi.org/10.1177/0160017603253789"
}

@book{Kemeny1967,
    Author = "Kemeny, John G. and Snell, James Laurie",
    Date-Added = "2018-06-12 22:44:29 +0000",
    Date-Modified = "2018-06-12 22:44:40 +0000",
    Keywords = "markov chain, ergodic.py, mobility",
    Publisher = "Van Nostrand",
    Title = "Finite markov chains",
    Year = "1967",
    Bdsk-File-1 = " "
}

@article{Rey2001,
    Author = "Rey, Sergio J.",
    Date-Added = "2018-06-12 21:11:59 +0000",
    Date-Modified = "2018-06-12 21:13:29 +0000",
    Doi = "10.1111/j.1538-4632.2001.tb00444.x",
    Journal = "Geographical Analysis",
    Number = "3",
    Pages = "195--214",
    Publisher = "Wiley Online Library",
    Title = "Spatial empirics for economic growth and convergence",
    Url = "https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.2001.tb00444.x",
    Volume = "33",
    Year = "2001",
    Bdsk-Url-1 = "https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1538-4632.2001.tb00444.x",
    Bdsk-Url-2 = "https://doi.org/10.1111/j.1538-4632.2001.tb00444.x"
}

@Article{rey_interregional_2010,
    Author = "Rey, Sergio J. and Sastr\'e-Guti\'errez, Myrna L",
    Title = "Interregional inequality dynamics in {Mexico}",
    Journal = "Spatial Economic Analysis",
    Year = "2010",
    Number = "3",
    Pages = "277--298",
    Volume = "5"
}

@Article{rey_comparative_2015,
    Author = "Rey, Sergio J. and Sastr\'e Guti\'errez, Myrna L.",
    Title = "Comparative spatial inequality dynamics: The case of {Mexico and the United States}",
    Journal = "Applied Geography",
    Year = "2015",
    Month = "Jul",
    Pages = "70--80",
    Volume = "61",
    Bdsk-url-1 = "http://dx.doi.org/10.1016/j.apgeog.2015.01.012",
    Doi = "10.1016/j.apgeog.2015.01.012",
    ISSN = "0143-6228",
    Publisher = "Elsevier BV",
    Url = "http://dx.doi.org/10.1016/j.apgeog.2015.01.012"
}

@Article{Rey_2013_sea,
    Author = "Rey, Sergio J and Smith, Richard J",
    Title = "A spatial decomposition of the {Gini} coefficient",
    Journal = "Letters in Spatial and Resource Sciences",
    Year = "2013",
    Pages = "55--70",
    Volume = "6",
    Keywords = "gini, inequality, space-time"
}

@article{VanLieshout1996,
    Author = "Lieshout, M. N. M. and Baddeley, A. J.",
    Abstract = "The strength and range of interpoint interactions in a spatial point process can be quantified by the function J = (1 - G)/(1 - F), where G is the nearest-neighbour distance distribution function and F the empty space function of the process. J(r) is identically equal to 1 for a Poisson process; values of J(r) smaller or larger than 1 indicate clustering or regularity, respectively. We show that, for a large class of point processes, J(r) is constant for distances r greater than the range of spatial interaction. Hence both the range and type of interaction can be inferred from J without parametric model assumptions. It is also possible to evaluate J(r) explicitly for many point process models, so that J is also useful for parameter estimation. Various properties are derived, including the fact that the J function of the superposition of independent point processes is a weighted mean of the J functions of the individual processes. Estimators of J can be constructed from standard estimators of F and G. We compute estimates of J for several standard point pattern datasets and implement a Monte Carlo test for complete spatial randomness.",
    Date-Added = "2018-11-12 17:23:48 -0800",
    Date-Modified = "2018-11-12 17:24:22 -0800",
    Doi = "10.1111/j.1467-9574.1996.tb01501.x",
    Journal = "Statistica Neerlandica",
    Keywords = "clustering, empty space function, J-statistic, Monte Carlo inference, nearest-neighbour distance distribution, Nguyen-Zessin formula, point process, spatial interaction, spatial statistics, regularity",
    Number = "3",
    Pages = "344-361",
    Title = "A nonparametric measure of spatial interaction in point patterns",
    Volume = "50",
    Year = "1996",
    Bdsk-Url-1 = "https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9574.1996.tb01501.x",
    Bdsk-Url-2 = "https://doi.org/10.1111/j.1467-9574.1996.tb01501.x"
}

@article{Dijkstra1959a,
    author = "Dijkstra, E. W.",
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}

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    Title = "An evaluation of sampling and full enumeration strategies for {Fisher Jenks} classification in big data settings",
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}

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@book{Anselin1988,
    Author = "Anselin, Luc",
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}

@article{Arraiz2010,
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    Keywords = "spatial econometrics,sigma convergence",
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    Title = "A Spatial {{Cliff}}-{{Ord}}-Type Model with Heteroskedastic Innovations: {{Small}} and Large Sample Results",
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    Year = "2010",
    Bdsk-Url-1 = "https://doi.org/10.1111/j.1467-9787.2009.00618.x"
}

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    Author = "Belsley, David A and Kuh, Edwin and Welsch, Roy E",
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@article{yu:2019,
    Author = "Yu, Hanchen and Fotheringham, Alexander Stewart and Li, Ziqi and Oshan, Taylor and Kang, Wei and Wolf, Levi John",
    Abstract = "A recent paper expands the well-known geographically weighted regression (GWR) framework significantly by allowing the bandwidth or smoothing factor in GWR to be derived separately for each covariate in the model---a framework referred to as multiscale GWR (MGWR). However, one limitation of the MGWR framework is that, until now, no inference about the local parameter estimates was possible. Formally, the so-called ``hat matrix,'' which projects the observed response vector into the predicted response vector, was available in GWR but not in MGWR. This paper addresses this limitation by reframing GWR as a Generalized Additive Model, extending this framework to MGWR and then deriving standard errors for the local parameters in MGWR. In addition, we also demonstrate how the effective number of parameters can be obtained for the overall fit of an MGWR model and for each of the covariates within the model. This statistic is essential for comparing model fit between MGWR, GWR, and traditional global models, as well as for adjusting multiple hypothesis tests. We demonstrate these advances to the MGWR framework with both a simulated data set and a real-world data set and provide a link to new software for MGWR (MGWR1.0) which includes the novel inferential framework for MGWR described here.",
    Date-Added = "2019-06-29 12:32:28 -0700",
    Date-Modified = "2019-06-29 12:34:19 -0700",
    Doi = "10.1111/gean.12189",
    Eprint = "https://onlinelibrary.wiley.com/doi/pdf/10.1111/gean.12189",
    Journal = "Geographical Analysis",
    Number = "0",
    Title = "Inference in Multiscale Geographically Weighted Regression",
    Url = "https://onlinelibrary.wiley.com/doi/abs/10.1111/gean.12189",
    Volume = "0",
    Year = "2019",
    Bdsk-File-1 = " ",
    Bdsk-Url-1 = "https://doi.org/10.1111/gean.12189"
}

@article{harris_use_2010,
    Author = "Harris, P. and Fotheringham, A. S. and Crespo, R. and Charlton, M.",
    Abstract = "Increasingly, the geographically weighted regression (GWR) model is being used for spatial prediction rather than for inference. Our study compares GWR as a predictor to (a) its global counterpart of multiple linear regression (MLR); (b) traditional geostatistical models such as ordinary kriging (OK) and universal kriging (UK), with MLR as a mean component; and (c) hybrids, where kriging models are specified with GWR as a mean component. For this purpose, we test the performance of each model on data simulated with differing levels of spatial heterogeneity (with respect to data relationships in the mean process) and spatial autocorrelation (in the residual process). Our results demonstrate that kriging (in a UK form) should be the preferred predictor, reflecting its optimal statistical properties. However the GWR-kriging hybrids perform with merit and, as such, a predictor of this form may provide a worthy alternative to UK for particular (non-stationary relationship) situations when UK models cannot be reliably calibrated. GWR predictors tend to perform more poorly than their more complex GWR-kriging counterparts, but both GWR-based models are useful in that they provide extra information on the spatial processes generating the data that are being predicted.",
    Date-Added = "2018-09-26 11:37:20 -0700",
    Date-Modified = "2018-09-26 11:37:20 -0700",
    Doi = "10.1007/s11004-010-9284-7",
    File = "Full Text PDF:/Users/toshan/Library/Application Support/Zotero/Profiles/hc1axs85.default/zotero/storage/TI8VUNW5/Harris et al. - 2010 - The Use of Geographically Weighted Regression for .pdf:application/pdf;Snapshot:/Users/toshan/Library/Application Support/Zotero/Profiles/hc1axs85.default/zotero/storage/BD5EGNCN/10.html:text/html",
    Issn = "1874-8961, 1874-8953",
    Journal = "Mathematical Geosciences",
    Keywords = "Geotechnical Engineering, GWR, Hydrogeology, Kriging, Mathematical Applications in Earth Sciences, Relationship heterogeneity, Relationship nonstationarity, Spatial interpolation, Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences",
    Language = "en",
    Month = "June",
    Number = "6",
    Pages = "657--680",
    Shorttitle = "The {Use} of {Geographically} {Weighted} {Regression} for {Spatial} {Prediction}",
    Title = "The {Use} of {Geographically} {Weighted} {Regression} for {Spatial} {Prediction}: {An} {Evaluation} of {Models} {Using} {Simulated} {Data} {Sets}",
    Url = "http://link.springer.com/article/10.1007/s11004-010-9284-7",
    Urldate = "2015-12-14",
    Volume = "42",
    Year = "2010",
    Bdsk-Url-1 = "http://link.springer.com/article/10.1007/s11004-010-9284-7",
    Bdsk-Url-2 = "https://doi.org/10.1007/s11004-010-9284-7"
}

@book{belsey_regression_1980,
    Author = "Belsey, D. A. and Kuh, E. and Welsch, R. E.",
    Address = "New York",
    Date-Added = "2018-09-26 11:37:05 -0700",
    Date-Modified = "2018-09-26 11:37:05 -0700",
    Publisher = "Wiley",
    Title = "Regression {Diagnostics}: {Identifying} {Influential} {Data} and {Sources} of {Collinearity}",
    Year = "1980"
}

@book{fotheringham_geographically_2002,
    Author = "Fotheringham, A. Stewart and Brunsdon, Chris and Charlton, Martin",
    Abstract = "Geographical Weighted Regression (GWR) is a new local modelling technique for analysing spatial analysis. This technique allows local as opposed to global models of relationships to be measured and mapped. This is the first and only book on this technique, offering comprehensive coverage on this new 'hot' topic in spatial analysis. * Provides step-by-step examples of how to use the GWR model using data sets and examples on issues such as house price determinants, educational attainment levels and school performance statistics * Contains a broad discussion of and basic concepts on GWR through to ideas on statistical inference for GWR models * uniquely features accompanying author-written software that allows users to undertake sophisticated and complex forms of GWR within a user-friendly, Windows-based, front-end (see book for details).",
    Date-Added = "2018-09-26 11:36:59 -0700",
    Date-Modified = "2018-09-26 11:36:59 -0700",
    Isbn = "978-0-470-85525-6",
    Keywords = "Science / Earth Sciences / Geography",
    Language = "en",
    Month = "February",
    Publisher = "John Wiley \\& Sons",
    Shorttitle = "Geographically {Weighted} {Regression}",
    Title = "Geographically {Weighted} {Regression}: {The} {Analysis} of {Spatially} {Varying} {Relationships}",
    Year = "2002"
}

@article{wheeler_diagnostic_2007,
    Author = "Wheeler, David C.",
    Abstract = "Geographically weighted regression (GWR) is drawing attention as a statistical method to estimate regression models with spatially varying relationships between explanatory variables and a response variable. Local collinearity in weighted explanatory variables leads to GWR coefficient estimates that are correlated locally and across space, have inflated variances, and are at times counterintuitive and contradictory in sign to the global regression estimates. The presence of local collinearity in the absence of global collinearity necessitates the use of diagnostic tools in the local regression model building process to highlight areas in which the results are not reliable for statistical inference. The method of ridge regression can also be integrated into the GWR framework to constrain and stabilize regression coefficients and lower prediction error. This paper presents numerous diagnostic tools and ridge regression in GWR and demonstrates the utility of these techniques with an example using the Columbus crime dataset.",
    Date-Added = "2018-09-26 11:36:47 -0700",
    Date-Modified = "2018-09-26 11:36:47 -0700",
    Doi = "10.1068/a38325",
    File = "Full Text PDF:/Users/toshan/Library/Application Support/Zotero/Profiles/hc1axs85.default/zotero/storage/AMG2X653/Wheeler - 2007 - Diagnostic Tools and a Remedial Method for Colline.pdf:application/pdf;Snapshot:/Users/toshan/Library/Application Support/Zotero/Profiles/hc1axs85.default/zotero/storage/DTMFIAI7/2464.html:text/html",
    Issn = "0308-518X, 1472-3409",
    Journal = "Environment and Planning A",
    Language = "en",
    Month = "October",
    Number = "10",
    Pages = "2464--2481",
    Title = "Diagnostic {Tools} and a {Remedial} {Method} for {Collinearity} in {Geographically} {Weighted} {Regression}",
    Url = "http://epn.sagepub.com/content/39/10/2464",
    Urldate = "2015-10-27",
    Volume = "39",
    Year = "2007",
    Bdsk-Url-1 = "http://epn.sagepub.com/content/39/10/2464",
    Bdsk-Url-2 = "https://doi.org/10.1068/a38325"
}

@article{oshan_comparison_2017,
    Author = "Oshan, Taylor M. and Fotheringham, A. Stewart",
    Date-Added = "2018-09-26 11:36:35 -0700",
    Date-Modified = "2018-09-26 11:36:35 -0700",
    Doi = "10.1111/gean.12133",
    File = "Oshan\_et\_al-2017-Geographical\_Analysis.pdf:/Users/toshan/Library/Application Support/Zotero/Profiles/hc1axs85.default/zotero/storage/I5JFAIZQ/Oshan\_et\_al-2017-Geographical\_Analysis.pdf:application/pdf",
    Issn = "00167363",
    Journal = "Geographical Analysis",
    Language = "en",
    Month = "June",
    Shorttitle = "A {Comparison} of {Spatially} {Varying} {Regression} {Coefficient} {Estimates} {Using} {Geographically} {Weighted} and {Spatial}-{Filter}-{Based} {Techniques}",
    Title = "A {Comparison} of {Spatially} {Varying} {Regression} {Coefficient} {Estimates} {Using} {Geographically} {Weighted} and {Spatial}-{Filter}-{Based} {Techniques}: {A} {Comparison} of {Spatially} {Varying} {Regression}",
    Url = "http://doi.wiley.com/10.1111/gean.12133",
    Urldate = "2017-07-01",
    Year = "2017",
    Bdsk-Url-1 = "http://doi.wiley.com/10.1111/gean.12133",
    Bdsk-Url-2 = "https://doi.org/10.1111/gean.12133"
}

@article{nakaya2005geographically,
    Author = "Nakaya, T and Fotheringham, AS and Brunsdon, Chris and Charlton, Martin",
    Date-Added = "2018-09-26 11:04:53 -0700",
    Date-Modified = "2018-09-26 11:04:53 -0700",
    Journal = "Statistics in Medicine",
    Number = "17",
    Pages = "2695--2717",
    Publisher = "Wiley Online Library",
    Title = "Geographically weighted Poisson regression for disease association mapping",
    Volume = "24",
    Year = "2005",
    Bdsk-File-1 = " "
}

@article{brunsdon2008geographically,
    Author = "Brunsdon, Chris and Fotheringham, A Stewart and Charlton, Martin",
    Date-Added = "2018-09-26 11:04:50 -0700",
    Date-Modified = "2018-09-26 11:04:50 -0700",
    Journal = "Encyclopedia of Geographic Information Science",
    Pages = "558",
    Publisher = "SAGE Publications",
    Title = "Geographically weighted regression: a method for exploring spatial nonstationarity",
    Year = "2008",
    Bdsk-File-1 = " "
}

@article{Fotheringham2016,
    Author = "Fotheringham, A. Stewart and Oshan, Taylor M.",
    Abstract = "Geographically weighted regression (GWR) extends the familiar regression framework by estimating a set of parameters for any number of locations within a study area, rather than producing a single parameter estimate for each relationship specified in the model. Recent literature has suggested that GWR is highly susceptible to the effects of multicollinearity between explanatory variables and has proposed a series of local measures of multicollinearity as an indicator of potential problems. In this paper, we employ a controlled simulation to demonstrate that GWR is in fact very robust to the effects of multicollinearity. Consequently, the contention that GWR is highly susceptible to multicollinearity issues needs rethinking.",
    Date-Added = "2018-09-26 11:04:45 -0700",
    Date-Modified = "2018-09-26 11:04:45 -0700",
    Doi = "10.1007/s10109-016-0239-5",
    Issn = "1435-5949",
    Journal = "Journal of Geographical Systems",
    Keywords = "GWR",
    Number = "4",
    Pages = "303--329",
    Title = "Geographically weighted regression and multicollinearity: dispelling the myth",
    Url = "http://dx.doi.org/10.1007/s10109-016-0239-5",
    Volume = "18",
    Year = "2016",
    Bdsk-File-1 = " ",
    Bdsk-Url-1 = "http://dx.doi.org/10.1007/s10109-016-0239-5"
}

@article{fotheringham1999local,
    Author = "Fotheringham, A Stewart and Brunsdon, Chris",
    Date-Added = "2018-09-26 11:04:32 -0700",
    Date-Modified = "2018-09-26 11:04:32 -0700",
    Journal = "Geographical Analysis",
    Number = "4",
    Pages = "340--358",
    Publisher = "Wiley Online Library",
    Title = "Local forms of spatial analysis",
    Volume = "31",
    Year = "1999",
    Bdsk-File-1 = " "
}

@article{Fotheringham:2017,
    Author = "Fotheringham, A. Stewart and Yang, Wenbai and Kang, Wei",
    Date-Added = "2018-09-26 11:04:20 -0700",
    Date-Modified = "2018-09-26 11:04:20 -0700",
    Doi = "10.1080/24694452.2017.1352480",
    Eprint = "http://dx.doi.org/10.1080/24694452.2017.1352480",
    Journal = "Annals of the American Association of Geographers",
    Number = "6",
    Pages = "1247-1265",
    Publisher = "Taylor \& Francis",
    Title = "Multiscale Geographically Weighted Regression (MGWR)",
    Url = "http://dx.doi.org/10.1080/24694452.2017.1352480",
    Volume = "107",
    Year = "2017",
    Bdsk-File-1 = " "
}

@article{brunsdon:1999,
    Author = "Brunsdon, Chris and Fotheringham, A Stewart and Charlton, Martin",
    Date-Added = "2018-09-26 11:04:11 -0700",
    Date-Modified = "2018-09-26 11:04:11 -0700",
    Journal = "Journal of Regional Science",
    Number = "3",
    Pages = "497--524",
    Publisher = "Wiley Online Library",
    Title = "Some notes on parametric significance tests for geographically weighted regression",
    Volume = "39",
    Year = "1999",
    Bdsk-File-1 = " "
}

@article{Silva:2016,
    Author = "da Silva, Alan Ricardo and Fotheringham, A. Stewart",
    Abstract = "This article describes the problem of multiple testing within a Geographically Weighted Regression framework and presents a possible solution to the problem which is based on a family-wise error rate for dependent processes. We compare the solution presented here to other solutions such as the Bonferroni correction and the Byrne, Charlton, and Fotheringham proposal which is based on the Benjamini and Hochberg False Discovery Rate. We conclude that our proposed correction is superior to others and that generally some correction in the conventional t-test is necessary to avoid false positives in GWR.",
    Date-Added = "2018-09-26 11:02:38 -0700",
    Date-Modified = "2018-09-26 11:03:28 -0700",
    Doi = "10.1111/gean.12084",
    Eprint = "https://onlinelibrary.wiley.com/doi/pdf/10.1111/gean.12084",
    Journal = "Geographical Analysis",
    Number = "3",
    Pages = "233-247",
    Title = "The Multiple Testing Issue in Geographically Weighted Regression",
    Url = "https://onlinelibrary.wiley.com/doi/abs/10.1111/gean.12084",
    Volume = "48",
    Year = "2016",
    Bdsk-Url-1 = "https://onlinelibrary.wiley.com/doi/abs/10.1111/gean.12084",
    Bdsk-Url-2 = "https://doi.org/10.1111/gean.12084"
}
